Quantum Theory of ۳+۱ Gravity and Dark Matter: A New Formulation of The Gupta-Feynman Based Quantum Field Theory of ۳+۱ Einstein Gravity

Gravitons and axions play an important role with regards to dark matter. Here, by appeal to developments of Gupta and Feynman and using a novel mathematical theory based on Ultrahyperfunctions [۱] , we are able to provide an exact, quantum relativistic expression for the gravitons and axions self-energies. For a complete explanation of Ultrahyperfunctions and their uses in Quantum Field Theory see [۲]. Ultrahyperfunctions (UHF) are in the most cases the generalization and extension to the complex plane of Schwartz ’tempered distributions. For example, in our book [۲] and in the references [۳, ۴, ۵, ۶] you can find a large number of examples of Ultrahyperfunctions. This manuscript is an application to Einstein’s Gravity and Dark Matter (EG) of the mathematical theory developed by Bollini et al [۳, ۴, ۵, ۶] and continued for more than ۲۵ years by one of the authors of this paper. We will quantize EG using the most general quantization approach, the Schwinger-Feynman variational principle [۱۵], which is more appropriate and rigorous than the popular functional integral method (FIM). FIM is not applicable here because our Lagrangian contains derivative couplings. We use the Einstein Lagrangian as obtained by Gupta [۱۶, ۱۷, ۱۸], but we added a new constraint to the theory. Thus the problem of lack of unitarity for the S matrix that appears in the procedures of Gupta and Feynman disappear Furthermore, we considerably simplify the handling of constraints, eliminating the need to appeal to ghosts for guarantying unitarity of the theory. Our theory is obviously non-renormalizable. However, this inconvenience is solved by resorting to the theory developed by Bollini et al. [۳, ۴, ۵, ۶] This theory is based on the thesis of Alexander Grothendieck [۱۹] and on the theory of Ultrahyperfunctions Based on these papers, a complete theory has been constructed for ۲۵ years that is able to quantize non-renormalizable Field Theories (FT).Because we are using a Gupta-Feynman based EG Lagrangian and to the new mathematical theory we have avoided the use of ghosts, as we have already mentioned, to obtain a unitary QFT of EG Moreover the self-energy of the graviton changes its mass and propagator upon interaction with the axion. The mass of the graviton can increase and the bare propagator changes to the dressed propagator. This phenomenon is measurable, but very difficult to detect since the bare mass of the graviton is zero and that of the axion is extremely small. Also, for the first time in the literature, we give explicit formulas for the self-energy of the graviton, interacting and non-interacting with axions. Also, for the first time in the literature, we present ۱۷ graphs corresponding to those self-energies.

کلیدواژه ها:

Quantum Field Theory ، Einstein gravity ، Axions ، Dark matter ، Non-renormalizable theories ، Unitarity. Ultrahyperfunctions